Is there any system that generates website designs?
The closest research that I am aware of is in artificial intelligence designed user interfaces(1,2).

The scope of which vary tremendously. Some teams are trying to generate UIs based on some user-defined parameters, others are trying to generate based off of images(as in the second link). I think part of the reason research is focused here is two-fold: One, we aren't very good at generating UIs automatically yet, and 2, it is an exponentially harder problem to develop a backend that has integrated calls and features required for a fully-featured website or application.

With that being said, one could adapt this research into things like one-page sites or other simple implementations that might bear fruit.